Tools

"... An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonst ..."

An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.

"... Abstract. In this note, we establish some results which suggest a possible solution to the problem of finding the right constructive notion of compactness in the context of a not–necessarily–uniform apartness space. ..."

Abstract. In this note, we establish some results which suggest a possible solution to the problem of finding the right constructive notion of compactness in the context of a not–necessarily–uniform apartness space.

"... Abstract: In this paper we study various properties of complements of sets and the Efremovič separation property in a symmetric pre–apartness space. Key Words: Pre–apartness spaces, Efremovič property Category: F.4.1 The constructive theory of apartness 2 (point–set and set–set) has been developed w ..."

Abstract: In this paper we study various properties of complements of sets and the Efremovič separation property in a symmetric pre–apartness space. Key Words: Pre–apartness spaces, Efremovič property Category: F.4.1 The constructive theory of apartness 2 (point–set and set–set) has been developed within the framework of Bishop’s constructive mathematics BISH [1, 2, 3, 13] in a series of papers over the past five years [17, 5, 12, 14, 7]. In this paper we derive some basic properties of complements of sets in pre–apartness spaces and discuss a strong separation property. Our starting point is a set X equipped with an inequality relation applicable to points of X, and a symmetric relation ⊲ ⊳ applicable to subsets of X. The inequality satisfies two simple properties x � = y ⇒ y � = x x � = y ⇒¬(x = y). Forapointx of X we write x⊲⊳Sas shorthand for {x} ⊲ ⊳ S. There are three notions of complement applicable to a subset S of X: – the logical complement – the complement – and the apartness complement ¬S = {x ∈ X: x/ ∈ S}, ∼ S = {x ∈ X: ∀s ∈ S (x � = s)},